Crossing Numbers of Random Two-Bridge Knots

TL;DR

This paper derives a closed-form distribution for crossing numbers of random two-bridge knots modeled via billiard table diagrams, confirming exponential decay of the probability for any specific knot as the model size increases.

Contribution

It provides a closed formula for crossing number distribution and proves exponential decay of individual knot probabilities in the random model.

Findings

Distribution of crossing numbers is explicitly characterized.

Probability of any specific knot decreases exponentially with model size.

Confirms a previously conjectured decay behavior.

Abstract

In a previous work, the first and third authors studied a random knot model for all two-bridge knots using billiard table diagrams. Here we present a closed formula for the distribution of the crossing numbers of such random knots. We also show that the probability of any given knot appearing in this model decays to zero at an exponential rate as the length of the billiard table goes to infinity. This confirms a conjecture from the previous work.